Letters in Mathematical Physics

, Volume 106, Issue 7, pp 913–923 | Cite as

Incompatibility of Time-Dependent Bogoliubov–de-Gennes and Ginzburg–Landau Equations

  • Rupert L. Frank
  • Christian Hainzl
  • Benjamin Schlein
  • Robert Seiringer
Open Access


We study the time-dependent Bogoliubov–de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg–Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior.


superconductivity quasi-free states critical temperature BCS theory 

Mathematics Subject Classification

82D50 46N50 


  1. 1.
    Ginzburg V.L., Landau L.D.: On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064 (1950)Google Scholar
  2. 2.
    Bardeen J., Cooper L., Schrieffer J.: Theory of superconductivity. Phys. Rev. 108, 1175 (1957)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Gor’kov L.P.: Microscopic derivation of the Ginzburg–Landau equations in the theory of superconductivity. ZH. Eksp. Teor. Fiz. 36, 1918 (1959)MATHGoogle Scholar
  4. 4.
    Eilenberger G.: Ableitung verallgemeinerter Ginzburg–Landau-Gleichungen für reine Supraleiter aus einem Variationsprinzip. Z. Phys. 182, 427 (1965)ADSCrossRefMATHGoogle Scholar
  5. 5.
    De Gennes, P.G.: Superconductivity of metals and alloys, Advanced Books Classics Series. Westview Press (1999)Google Scholar
  6. 6.
    Frank R.L., Hainzl C., Seiringer R., Solovej J.P.: Microscopic derivation of the Ginzburg–Landau theory. J. Am. Math. Soc. 25, 667 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Frank R.L., Hainzl C., Seiringer R., Solovej J.P.: The external field dependence of the BCS critical temperature. Commun. Math. Phys. 342, 189 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Stephen M.J., Suhl H.: Weak time dependence in pure superconductors. Phys. Rev. Lett. 13, 797 (1964)ADSCrossRefMATHGoogle Scholar
  9. 9.
    Abrahams E., Tsuneto T.: Time variation of the Ginzburg–Landau order parameter. Phys. Rev. 152, 416 (1966)ADSCrossRefGoogle Scholar
  10. 10.
    Schmidt H.: The onset of superconductivity in the time dependent Ginzburg–Landau theory. Z. Phys. 216, 336 (1968)ADSCrossRefGoogle Scholar
  11. 11.
    Gor’kov L.P., Eliashberg G.M.: Generalization of Ginzburg–Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Sov. Phys. JETP. 27, 328 (1968)ADSGoogle Scholar
  12. 12.
    Cyrot M.: Ginzburg–Landau theory for superconductors. Rep. Prog. Phys. 36, 103 (1973)ADSCrossRefGoogle Scholar
  13. 13.
    Sá de Melo C.A.R., Randeria M., Engelbrecht J.R.: Crossover from BCS to Bose superconductivity: transition temperature and time-dependent Ginzburg–Landau theory. Phys. Rev. Lett. 71, 3202 (1993)Google Scholar
  14. 14.
    Randeria, M.: Crossover from BCS theory to Bose–Einstein condensation. In: Griffin, A., Snoke, D.W., Stringari, S., (eds.) Bose-Einstein condensation, Cambridge University Press, pp. 355–392 (1996)Google Scholar
  15. 15.
    Gor’kov L.P.: On the energy spectrum of superconductors. Sov. Phys. JETP. 34, 505 (1958)MathSciNetMATHGoogle Scholar
  16. 16.
    Hainzl, C., Seyrich, J.: Comparing the full time-dependent BCS equation to its linear approximation: a numerical investigation. Eur. Phys. J. (To appear). arXiv:1504.05881
  17. 17.
    Hainzl C., Hamza E., Seiringer R., Solovej J.P.: The BCS functional for general pair interactions. Commun. Math. Phys. 281, 349 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Frank R.L., Hainzl C., Naboko S., Seiringer R.: The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17, 559 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hainzl C., Seiringer R.: Critical temperature and energy gap for the BCS equation. Phys. Rev. B. 77, 184517 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    Frank, R.L., Lemm, M.: Multi-component Ginzburg–Landau theory: microscopic derivation and examples. Ann. H. Poincaré. arXiv:1504.07306
  21. 21.
    Zwerger, W. (ed.): The BCS-BEC crossover and the unitary Fermi gas, Lecture notes in physics, vol. 836. Springer (2012)Google Scholar
  22. 22.
    Leggett, A.J.: In: Pekalski, A., Przystawa, J. (eds.): Modern trends in the theory of condensed matter, Lecture notes in physics, vol. 115, pp. 13–27. Springer (1980)Google Scholar
  23. 23.
    Noziéres P., Schmitt-Rink S.: Bose condensation in an attractive fermion gas: from weak to strong coupling superconductivity. J. Low Temp. Phys. 59, 195 (1985)ADSCrossRefGoogle Scholar
  24. 24.
    Drechsler M., Zwerger W.: Crossover from BCS-superconductivity to Bose-condensation. Ann. Phys. 504, 15 (1990)CrossRefGoogle Scholar
  25. 25.
    Pieri P., Strinati G.C.: Derivation of the Gross-Pitaevskii equation for condensed bosons from the Bogoliubov–de-Gennes equations for superfluid fermions. Phys. Rev. Lett. 91, 030401 (2003)ADSCrossRefGoogle Scholar
  26. 26.
    Hainzl C., Seiringer R.: Low density limit of BCS theory and Bose-Einstein condensation of fermion pairs. Lett. Math. Phys. 100, 119 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hainzl C., Schlein B.: Dynamics of Bose–Einstein condensates of fermion pairs in the low density limit of BCS theory. J. Funct. Anal. 265, 399 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Andreev A.F.: The thermal conductivity of the intermediate state in superconductors. Sov. Phys. JETP. 19, 1228 (1964)Google Scholar
  29. 29.
    Kümmel R.: Dynamics of current flow through the phase-boundary between a normal and a superconducting region. Z. Phys. 218, 472 (1969)ADSCrossRefGoogle Scholar
  30. 30.
    Ambegaokar, V.: In: Parks, R.D. (ed.) Superconductivity. Dekker, New York (1969)Google Scholar
  31. 31.
    Hunziker W.: Resonances, metastable states and exponential decay laws in perturbation theory. Commun. Math. Phys. 132, 177 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Rupert L. Frank
    • 1
  • Christian Hainzl
    • 2
  • Benjamin Schlein
    • 3
  • Robert Seiringer
    • 4
  1. 1.Mathematics 253-37CaltechPasadenaUSA
  2. 2.Mathematisches InstitutUniversität TübingenTübingenGermany
  3. 3.Institute of MathematicsUniversity of ZürichZurichSwitzerland
  4. 4.Institute of Science and Technology Austria (IST Austria)KlosterneuburgAustria

Personalised recommendations